Thin Spectra and Singular Continuous Spectral Measures for Limit-Periodic Jacobi Matrices

TL;DR

This paper studies the spectral properties of limit-periodic Jacobi matrices, showing that generically their spectra are Cantor sets of zero measure with purely singular continuous spectral measures, and extends these results to multidimensional Laplacians.

Contribution

It demonstrates that for a residual set of limit-periodic Jacobi matrices, the spectrum is a zero-measure Cantor set with purely singular continuous measures, and provides methods to construct such spectra in various settings.

Findings

Residual set of matrices have zero-measure Cantor spectrum

Spectral measures are purely singular continuous

Examples of multidimensional Laplacians with zero-dimensional spectrum

Abstract

This paper investigates the spectral properties of Jacobi matrices with limit-periodic coefficients. We show that for a residual set of such matrices, the spectrum is a Cantor set of zero Lebesgue measure, and the spectral measures are purely singular continuous. For a dense set of limit-periodic Jacobi matrices we can strengthen the result and show that the spectrum is a Cantor set of zero lower box counting dimension, and hence in particular of zero Hausdorff dimension, while still retaining the singular continuity of the spectral type. We also show how results of this nature can be established by fixing the off-diagonal coefficients and varying only the diagonal coefficients, and, in a more restricted version, by fixing the diagonal coefficients to be zero and varying only the off-diagonal coefficients. We apply these results to produce examples of weighted Laplacians on the multidimensional integer lattice having purely singular continuous spectral type and zero-dimensional spectrum.